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What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation ...
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Propositional Equality, Identity Types and Computational Paths

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Apresentação no workshop "Logic and Applications: in honor to Francisco Miraglia on the occasion of his 70th birthday", IME-USP, 16–17 Set 2016.

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Propositional Equality, Identity Types and Computational Paths

  1. 1. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Propositional Equality, Identity Types and Computational Paths Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Logic and Applications: in honor to Francisco Miraglia on the occasion of his 70th birthday 16–17 Sept 2016 Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  2. 2. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Homotopy Type Theory Univalent Foundations of Mathematics Institute for Advanced Study, Princeton 484–600p. Open-source book: 27 main authors. 58 contributors Available on GitHub. Latest version September 3, 2016 Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  3. 3. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Homotopy Type Theory Univalent Foundations of Mathematics “Homotopy type theory is a new branch of mathematics that combines aspects of several different fields in a surprising way. It is based on a recently discovered connection between homotopy theory and type theory. Homotopy theory is an outgrowth of algebraic topology and homological algebra, with relationships to higher category theory; while type theory is a branch of mathematical logic and theoretical computer science. Although the connections between the two are currently the focus of intense investigation, it is increasingly clear that they are just the beginning of a subject that will take more time and more hard work to fully understand. It touches on topics as seemingly distant as the homotopy groups of spheres, the algorithms for type checking, and the definition of weak ∞-groupoids.” Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  4. 4. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Geometry and Logic Alexander Grothendieck Alexander Grothendieck b. 28 March 1928, Berlin, Prussia, Germany d. 13 November 2014 (aged 86), Saint-Girons, Ari`ege, France Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  5. 5. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Geometry and Logic Alexander Grothendieck . . . the study of n-truncated homotopy types (of semisimplicial sets, or of topological spaces) [should be] essentially equivalent to the study of so-called n-groupoids. . . . This is expected to be achieved by associating to any space (say) X its “fundamental n-groupoid” Πn(X).... The obvious idea is that 0-objects of Πn(X) should be the points of X, 1-objects should be “homotopies” or paths between points, 2-objects should be homotopies between 1-objects, etc. (Grothendieck 1983) Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  6. 6. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Equality in λ-Calculus: definitional vs. propositional Proofs of equality: Paths Church’s (1936) original paper: NB: equality as the reflexive, symmetric and transitive closure of 1-step contraction: rewriting paths. An algebra of paths (with α, β, η, µ, ν, ξ, ρ, σ, τ)? E.g. σ(σ(r)) = r, τ(τ(t, r), s) = τ(t, τ(r, s)).Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  7. 7. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Equality in λ-Calculus: definitional vs. propositional Proofs of equality: Paths Definition (Hindley & Seldin 2008) P is βη-equal or βη-convertible to Q (notation P =βη Q) iff Q is obtained from P by a finite (perhaps empty) series of β-contractions, η-contractions, reversed β-contractions, reversed η-contractions, or changes of bound variables. That is, P =β Q iff there exist P0, . . . , Pn (n ≥ 0) such that P0 ≡ P, Pn ≡ Q, (∀i ≤ n − 1) (Pi 1β Pi+1 or Pi+1 1β Pi or Pi 1η Pi+1 or Pi+1 1η Pi or Pi ≡α Pi+1). Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  8. 8. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Equality Sequences of contractions (λx.(λy.yx)(λw.zw))v 1η (λx.(λy.yx)z)v 1β (λy.yv)z 1β zv (λx.(λy.yx)(λw.zw))v 1β (λx.(λw.zw)x)v 1η (λx.zx)v 1β zv (λx.(λy.yx)(λw.zw))v 1β (λx.(λw.zw)x)v 1β (λw.zw)v 1η zv There is at least one sequence of contractions from the initial term to the final term. Thus, in the formal theory of λ-calculus, the term (λx.(λy.yx)(λw.zw))v is declared to be equal to zv. Now, some natural questions arise: 1 Are the sequences themselves normal? 2 What are the non-normal sequences? 3 How are the latter to be identified and (possibly) normalised? Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  9. 9. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Brouwer–Heyting–Kolmogorov Interpretation Proofs rather than truth-values a proof of the proposition: is given by: A ∧ B a proof of A and a proof of B A ∨ B a proof of A or a proof of B A → B a function that turns a proof of A into a proof of B ∀x.P(x) a function that turns an element a into a proof of P(a) ∃x.P(x) an element a (witness) and a proof of P(a) Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  10. 10. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Brouwer–Heyting–Kolmogorov Interpretation What is a proof of an equality statement? a proof of the proposition: is given by: t1 = t2 ? (Perhaps a path from t1 to t2?) What is the logical status of the symbol “=”? What would be a canonical/direct proof of t1 = t2? What is an equality between paths? What is an equality between homotopies (i.e., paths between paths)? Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  11. 11. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Geometry and Logic Vladimir Voevodsky “From an observation by Grothendieck: Formalism of higher equivalences (theory of grupoids) = Homotopy theory (theory of shapes up to a deformation) Combined with some other ideas it: leads to an encoding of mathematics in terms of the homotopy theory. Unlike the usual encodings in terms of the set theory this one respects equivalences.” Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  12. 12. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Type Theory and Equality Proposition vs Judgements In type theory, two main kinds of judgements: 1 x : A 2 x = y : A Via the so-called Curry-Howard interpretation, “x : A” can be read as “x is a proof of proposition A”. Also, “x = y : A” can be read as “x and y are (definitionally) equal proofs of proposition A”. What about the judgement of “p is a proof of the statement that x and y are equal elements of type A”? This is where the so-called Identity type comes into the picture: p : IdA(x, y) Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  13. 13. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Type Theory and its Derivations-as-Terms Interpretation Howard on Curry-Howard “ [de Bruijn] discovered the idea of derivations as terms, and the accompanying idea of formulae-as-types, on his own. (...) Martin-L¨of suggested that the derivations-as-terms idea would work particularly well in connection with Prawitz’s theory of natural deduction.” (W.Howard, Wadler’s Blog, 2014) Since the early 1990’s, our focus has been on insisting on the derivations-as-terms perspective, especially wrt to propositional equality. Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  14. 14. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Type Theory and Homotopy Theory Steve Awodey “The purpose of this informal survey article is to introduce the reader to a new and surprising connection between Geometry, Algebra, and Logic, which has recently come to light in the form of an interpretation of the constructive type theory of Per Martin-L¨of into homotopy theory, resulting in new examples of certain algebraic structures which are important in topology. This connection was discovered quite recently, and various aspects of it are now under active investigation by several researchers.” (“Type Theory and Homotopy Theory”, 2010.) Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  15. 15. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Algebraic Structure: Groupoids Steve Awodey “A groupoid is like a group, but with a partially-defined composition operation. Precisely, a groupoid can be defined as a category in which every arrow has an inverse. A group is thus a groupoid with only one object. Groupoids arise in topology as generalized fundamental groups, not tied to a choice of basepoint.” Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  16. 16. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P What is a proof of an equality statement? What is the formal counterpart of a proof of an equality? In talking about proofs of an equality statement, two dichotomies arise: 1 definitional equality versus propositional equality 2 intensional equality versus extensional equality First step on the formalisation of proofs of equality statements: Per Martin-L¨of’s Intuitionistic Type Theory (Log Coll ’73, published 1975) with the so-called Identity Type Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  17. 17. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Identity Types Identity Types - Topological and Categorical Structure Workshop, Uppsala, November 13–14, 2006 “The identity type, the type of proof objects for the fundamental propositional equality, is one of the most intriguing constructions of intensional dependent type theory (also known as Martin-L¨of type theory). Its complexity became apparent with the Hofmann–Streicher groupoid model of type theory. This model also hinted at some possible connections between type theory and homotopy theory and higher categories. Exploration of this connection is intended to be the main theme of the workshop.” Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  18. 18. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Identity Types Type Theory and Homotopy Theory Indeed, a whole new research avenue has since 2005 been explored by people like Vladimir Voevodsky and Steve Awodey in trying to make a bridge between type theory and homotopy theory, mainly via the groupoid structure exposed in the Hofmann–Streicher (1994) countermodel to the principle of Uniqueness of Identity Proofs (UIP). In Hofmann & Streicher’s own words, “We give a model of intensional Martin-L¨of type theory based on groupoids and fibrations of groupoids in which identity types may contain two distinct elements which are not even propositionally equal. This shows that the principle of uniqueness of identity proofs is not derivable in the syntax”. (“LICS ’94, pp. 208–212.) Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  19. 19. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Identity Types Identity Types as Topological Spaces According to B. van den Berg and R. Garner (“Topological and simplicial models of identity types”, ACM Transactions on Computational Logic, Jan 2012), “All of this work can be seen as an elaboration of the following basic idea: that in Martin-L¨of type theory, a type A is analogous to a topological space; elements a, b ∈ A to points of that space; and elements of an identity type p, q ∈ IdA(a, b) to paths or homotopies p, q : a → b in A.”. Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  20. 20. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Identity Types Identity Types as Topological Spaces From the Homotopy type theory collective book (2013): “In type theory, for every type A there is a (formerly somewhat mysterious) type IdA of identifications of two objects of A; in homotopy type theory, this is just the path space AI of all continuous maps I → A from the unit interval. In this way, a term p : IdA(a, b) represents a path p : a b in A.” Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  21. 21. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Identity Types: Iteration From Propositional to Predicate Logic and Beyond In the same aforementioned workshop, B. van den Berg in his contribution “Types as weak omega-categories” draws attention to the power of the identity type in the iterating types to form a globular set: “Fix a type X in a context Γ. Define a globular set as follows: A0 consists of the terms of type X in context Γ,modulo definitional equality; A1 consists of terms of the types Id(X; p; q) (in context Γ) for elements p, q in A0, modulo definitional equality; A2 consists of terms of well-formed types Id(Id(X; p; q); r; s) (in context Γ) for elements p, q in A0, r, s in A1, modulo definitional equality; etcetera...” Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  22. 22. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Identity Types: Iteration The homotopy interpretation Here is how we can see the connections between proofs of equality and homotopies: a, b : A p, q : IdA(a, b) α, β : IdIdA(a,b)(p, q) · · · : IdIdId... (· · · ) Now, consider the following interpretation: Types Spaces Terms Maps a : A Points a : 1 → A p : IdA(a, b) Paths p : a ⇒ b α : IdIdA(a,b)(p, q) Homotopies α : p q Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  23. 23. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Identity Types Univalent Foundations of Mathematics From Vladimir Voevodsky (IAS, Princeton) “Univalent Foundations: New Foundations of Mathematics”, Mar 26, 2014: “There were two main problems with the existing foundational systems which made them inadequate. Firstly, existing foundations of mathematics were based on the languages of Predicate Logic and languages of this class are too limited. Secondly, existing foundations could not be used to directly express statements about such objects as, for example, the ones that my work on 2-theories was about.” Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  24. 24. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Identity Types Univalent Foundations of Mathematics From Vladimir Voevodsky (IAS, Princeton) “Univalent Foundations: New Foundations of Mathematics”, Mar 26, 2014: “Univalent Foundations, like ZFC-based foundations and unlike category theory, is a complete foundational system, but it is very different from ZFC. To provide a format for comparison let me suppose that any foundation for mathematics adequate both for human reasoning and for computer verification should have the following three components.” Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  25. 25. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Identity Types Univalent Foundations of Mathematics From Vladimir Voevodsky (IAS, Princeton), “Univalent Foundations: New Foundations of Mathematics”, Mar 26, 2014: “The first component is a formal deduction system: a language and rules of manipulating sentences in this language that are purely formal, such that a record of such manipulations can be verified by a computer program. The second component is a structure that provides a meaning to the sentences of this language in terms of mental objects intuitively comprehensible to humans. The third component is a structure that enables humans to encode mathematical ideas in terms of the objects directly associated with the language.” Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  26. 26. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Propositional Equality Proofs of equality as (rewriting) computational paths What is a proof of an equality statement? In what sense it can be seen as a homotopy? Motivated by looking at equalities in type theory as arising from the existence of computational paths between two formal objects, it may be useful to review the role and the power of the notion of propositional equality as formalised in the so-called Curry–Howard functional interpretation. The main idea, i.e. proofs of equality statements as (reversible) sequences of rewrites, goes back to a paper entitled “Equality in labelled deductive systems and the functional interpretation of propositional equality”, , presented in Dec 1993 at the 9th Amsterdam Colloquium, and published in the proceedings in 1994. Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  27. 27. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Brouwer–Heyting–Kolmogorov Interpretation Proofs rather than truth-values a proof of the proposition: is given by: A ∧ B a proof of A and a proof of B A ∨ B a proof of A or a proof of B A → B a function that turns a proof of A into a proof of B ∀xD .P(x) a function that turns an element a into a proof of P(a) ∃xD .P(x) an element a (witness) and a proof of P(a) Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  28. 28. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Brouwer–Heyting–Kolmogorov Interpretation: Formally Canonical proofs rather than truth-values a proof of the proposition: has the canonical form of: A ∧ B p, q where p is a proof of A and q is a proof of B A ∨ B inl(p) where p is a proof of A or inr(q) where q is a proof of B (‘inl’ and ‘inr’ abbreviate ‘into the left/right disjunct’) A → B λx.b(x) where b(p) is a proof of B provided p is a proof of A ∀xD .P(x) Λx.f(x) where f(a) is a proof of P(a) provided a is an arbitrary individual chosen from the domain D ∃xD .P(x) f(a), a where a is a witness from the domain D, f(a) is a proof of P(a) Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  29. 29. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Brouwer–Heyting–Kolmogorov Interpretation What is a proof of an equality statement? a proof of the proposition: is given by: t1 = t2 ? (Perhaps a sequence of rewrites starting from t1 and ending in t2?) What is the logical status of the symbol “=”? What would be a canonical/direct proof of t1 = t2? Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  30. 30. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Statman’s Direct Computations Terms, Equations, Measure Definition (equations and systems of equations) Let us consider equations E between individual terms a, b, c, . . ., possibly containing function variables, and finite sets of equations S. Definition (measure) A function M from terms to non-negative integers is called a measure if M(a) ≤ M(b) implies M(c[a/x]) ≤ M(c[b/x]), and, whenever x occurs in c, M(a) ≤ M(c[a/x]). Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  31. 31. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Statman’s Direct Computations Kreisel–Tait’s calculus K Definition (calculus K) The calculus K of Kreisel and Tait consists of the axioms a = a and the rule of substituting equals for equals: (1) E[a/x] a . = b E[b/x] where a . = b is, ambiguously, a = b and b = a, together with the rules (2) sa = sb a = b (3) 0 = sa b = c (4) a = sn a b = c H will be the system consisting only of the rule (1) Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  32. 32. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Equality Sequences of contractions (λx.(λy.yx)(λw.zw))v η (λx.(λy.yx)z)v β (λy.yv)z β zv (λx.(λy.yx)(λw.zw))v β (λx.(λw.zw)x)v η (λx.zx)v β zv (λx.(λy.yx)(λw.zw))v β (λx.(λw.zw)x)v β (λw.zw)v η zv There is at least one sequence of contractions from the initial term to the final term. (In this case we have given three!) Thus, in the formal theory of λ-calculus, the term (λx.(λy.yx)(λw.zw))v is declared to be equal to zv. Now, some natural questions arise: 1 Are the sequences themselves normal? 2 Are there non-normal sequences? 3 If yes, how are the latter to be identified and (possibly) normalised? 4 What happens if general rules of equality are involved? Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  33. 33. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Equality Propositional equality Definition (Hindley & Seldin 2008) P is β-equal or β-convertible to Q (notation P =β Q) iff Q is obtained from P by a finite (perhaps empty) series of β-contractions and reversed β-contractions and changes of bound variables. That is, P =β Q iff there exist P0, . . . , Pn (n ≥ 0) such that P0 ≡ P, Pn ≡ Q, (∀i ≤ n − 1)(Pi 1β Pi+1 or Pi+1 1β Pi or Pi ≡α Pi+1). NB: equality with an existential force. NB: equality as the reflexive, symmetric and transitive closure of 1-step contraction: arising from rewriting Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  34. 34. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Direct Computation Equality: Existential Force and Rewriting Path The same happens with λβη-equality: Definition 7.5 (λβη-equality) (Hindley & Seldin 2008) The equality-relation determined by the theory λβη is called =βη; that is, we define M =βη N ⇔ λβη M = N. Note again that two terms are λβη-equal if there exists a proof of their equality in the theory of λβη-equality. Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  35. 35. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Equality Gentzen’s ND for propositional equality Remark In setting up a set of Gentzen’s ND-style rules for equality we need to account for: 1 definitional versus propositional equality; 2 there may be more than one normal proof of a certain equality statement; 3 given a (possibly non-normal) proof, the process of bringing it to a normal form should be finite and confluent. Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  36. 36. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Direct Computation Equality in Type Theory Martin-L¨of’s Intuitionistic Type Theory: Intensional (1975) Extensional (1982(?), 1984) Remark (Definitional vs. Propositional Equality) definitional, i.e. those equalities that are given as rewrite rules, orelse originate from general functional principles (e.g. β, η, ξ, µ, ν, etc.); propositional, i.e. the equalities that are supported (or otherwise) by an evidence (a sequence of substitutions and/or rewrites) Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  37. 37. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Direct Computation Definitional Equality Definition (Hindley & Seldin 2008) (α) λx.M = λy.[y/x]M (y /∈ FV(M)) (β) (λx.M)N = [N/x]M (η) (λx.Mx) = M (x /∈ FV(M)) (ξ) M = M λx.M = λx.M (µ) M = M NM = NM (ν) M = M MN = M N (ρ) M = M (σ) M = N N = M (τ) M = N N = P M = P Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  38. 38. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Direct Computation Intuitionistic Type Theory →-introduction [x : A] f(x) = g(x) : B λx.f(x) = λx.g(x) : A → B (ξ) →-elimination x = y : A g : A → B gx = gy : B (µ) x : A g = h : A → B gx = hx : B (ν) →-reduction a : A [x : A] b(x) : B (λx.b(x))a = b(a/x) : B (β) c : A → B λx.cx = c : A → B (η) Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  39. 39. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Direct Computations Lessons from Curry–Howard and Type Theory Harmonious combination of logic and λ-calculus; Proof terms as ‘record of deduction steps’, i.e. ‘deductions-as-terms’ Function symbols as first class citizens. Cp. ∃xP(x) [P(t)] C C with ∃xP(x) [t : D, f(t) : P(t)] g(f, t) : C ? : C in the term ‘?’ the variable f gets abstracted from, and this enforces a kind of generality to f, even if this is not brought to the ‘logical’ level.Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  40. 40. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Direct Computations Equality in Martin-L¨of’s Intensional Type Theory A type a : A b : A Idint A (a, b) type Idint -formation a : A r(a) : Idint A (a, a) Idint -introduction a = b : A r(a) : Idint A (a, b) Idint -introduction a : A b : A c : Idint A (a, b) [x:A] d(x):C(x,x,r(x)) [x:A,y:A,z:Idint A (x,y)] C(x,y,z) type J(c, d) : C(a, b, c) Idint -elimination a : A [x : A] d(x) : C(x, x, r(x)) [x : A, y : A, z : Idint A (x, y)] C(x, y, z) type J(r(a), d(x)) = d(a/x) : C(a, a, r(a)) Idint -equality Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  41. 41. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Direct Computations Equality in Martin-L¨of’s Extensional Type Theory A type a : A b : A Idext A (a, b) type Idext -formation a = b : A r : Idext A (a, b) Idext -introduction c : Idext A (a, b) a = b : A Idext -elimination c : Idext A (a, b) c = r : Idext A (a, b) Idext -equality Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  42. 42. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Direct Computations The missing entity Considering the lessons learned from Type Theory, the judgement of the form: a = b : A which says that a and b are equal elements from domain D, let us add a function symbol: a =s b : A where one is to read: a is equal to b because of ‘s’ (‘s’ being the rewrite reason); ‘s’ is a term denoting a sequence of equality identifiers (β, η, ξ, etc.), i.e. a composition of rewrites. In other words, ‘s’ is the computational path from a to b. (This formal entity is missing in both of Martin-L¨of’s formulations of Identity Types.) Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  43. 43. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Direct Computations Propositional Equality Id-introduction a =s b : A s(a, b) : IdA(a, b) Id-elimination m : IdA(a, b) [a =g b : A] h(g) : C J(m, λg.h(g)) : C Id-reduction a =s b : A s(a, b) : IdA(a, b) Id-intr [a =g b : A] h(g) : C J(s(a, b), λg.h(g)) : C Id-elim β [a =s b : A] h(s/g) : C Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  44. 44. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Direct Computations Propositional Equality: A Simple Example of a Proof By way of example, let us prove ΠxA ΠyA (IdA(x, y) → IdA(y, x)) [p : IdA(x, y)] [x =t y : A] y =σ(t) x : A (σ(t))(y, x) : IdA(y, x) J(p, λt(σ(t))(y, x)) : IdA(y, x) λp.J(p, λt(σ(t))(y, x)) : IdA(x, y) → IdA(y, x) λy.λp.J(p, λt(σ(t))(y, x)) : ΠyA(IdA(x, y) → IdA(y, x)) λx.λy.λp.J(p, λt(σ(t))(y, x)) : ΠxAΠyA(IdA(x, y) → IdA(y, x)) Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  45. 45. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Direct Computations Propositional Equality: The Groupoid Laws With the formulation of propositional equality that we have just defined, we can also prove that all elements of an identity type obey the groupoid laws, namely 1 Associativity 2 Existence of an identity element 3 Existence of inverses Also, the groupoid operation, i.e. composition of paths/sequences, is actually, partial, meaning that not all elements will be connected via a path. (The groupoid interpretation refutes the Uniqueness of Identity Proofs.) Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  46. 46. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Direct Computations Propositional Equality: The Uniqueness of Identity Proofs “We will call UIP (Uniqueness of Identity Proofs) the following property. If a1, a2 are objects of type A then for any proofs p and q of the proposition “a1 equals a2” there is another proof establishing equality of p and q. (...) Notice that in traditional logical formalism a principle like UIP cannot even be sensibly expressed as proofs cannot be referred to by terms of the object language and thus are not within the scope of propositional equality.” (Hofmann & Streicher 1996) Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  47. 47. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Direct Computations Normal form for the rewrite reasons Strategy: Analyse possibilities of redundancy Construct a rewriting system Prove termination and confluence Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  48. 48. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Direct Computations Normal form for the rewrite reasons Definition (equation) An equation in our LNDEQ is of the form: s =r t : A where s and t are terms, r is the identifier for the rewrite reason, and A is the type (formula). Definition (system of equations) A system of equations S is a set of equations: {s1 =r1 t1 : A1, . . . , sn =rn tn : An} where ri is the rewrite reason identifier for the ith equation in S. Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  49. 49. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Direct Computations Normal form for the rewrite reasons Definition (rewrite reason) Given a system of equations S and an equation s =r t : A, if S s =r t : A, i.e. there is a deduction/computation of the equation starting from the equations in S, then the rewrite reason r is built up from: (i) the constants for rewrite reasons: { ρ, σ, τ, β, η, ν, ξ, µ }; (ii) the ri’s; using the substitution operations: (iii) subL; (iv) subR; and the operations for building new rewrite reasons: (v) σ, τ, ξ, µ. Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  50. 50. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Direct Computations Normal form for the rewrite reasons Definition (general rules of equality) The general rules for equality (reflexivity, symmetry and transitivity) are defined as follows: x : A x =ρ x : A (reflexivity) x =t y : A y =σ(t) x : A (symmetry) x =t y : A y =u z : A x =τ(t,u) z : A (transitivity) Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  51. 51. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Direct Computations Normal form for the rewrite reasons Definition (subterm substitution) The rule of “subterm substitution” is split into two rules: x =r C[y] : A y =s u : A x =subL(r,s) C[u] : A x =r w : A C[w] =s u : A C[x] =subR(r,s) u : A where C[x] is the context in which the subterm x appears Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  52. 52. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Direct Computations Reductions Definition (reductions involving ρ and σ) x =ρ x : A x =σ(ρ) x : A sr x =ρ x : A x =r y : A y =σ(r) x : A x =σ(σ(r)) y : A ss x =r y : A Associated rewritings: σ(ρ) sr ρ σ(σ(r)) ss r Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  53. 53. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Direct Computations Reductions Definition (reductions involving τ) x=r y:D y=σ(r)x:D x=τ(r,σ(r))x:D tr x =ρ x : D y=σ(r)x:D x=r y:D y=τ(σ(r),r)y:D tsr y =ρ y : D u=r v:D v=ρv:D u=τ(r,ρ)v:D rrr u =r v : D u=ρu:D u=r v:D u=τ(ρ,r)v:D lrr u =r v : D Associated equations: τ(r, σ(r)) tr ρ, τ(σ(r), r) tsr ρ, τ(r, ρ) rrr r, τ(ρ, r) lrr r. Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  54. 54. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Direct Computations Reductions Definition βrewr -×-reduction x =r y : A z : B x, z =ξ1(r) y, z : A × B × -intr FST( x, z ) =µ1(ξ1(r)) FST( y, z ) : A × -elim mx2l1 x =r y : A Associated rewriting: µ1(ξ1(r)) mx2l1 r Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  55. 55. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Direct Computations Reductions Definition βrewr -×-reduction x =r x : A y =s z : B x, y =ξ∧(r,s) x , z : A × B × -intr FST( x, y ) =µ1(ξ∧(r,s)) FST( x , z ) : A × -elim mx2l2 x =r x : A Associated rewriting: µ1(ξ∧(r, s)) mx2l2 r Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  56. 56. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P Categorical Interpretation of Computational Paths Computational Paths form a Weak Category Theorem For each type A, computational paths induce a weak categorical structure Arw where: objects: terms a of the type A, i.e., a : A morphisms: a morphism (arrow) between terms a : A and b : A are arrows s : a → b such that s is a computational path between the terms, i.e., a =s b : A. Corollary Arw has a weak groupoidal structure. Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths
  57. 57. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of P The Functional Interpretation of Direct Computations Reductions Working papers: 1 Propositional equality, identity types, and direct computational paths Ruy J.G.B. de Queiroz, Anjolina G. de Oliveira http://arxiv.org/abs/1107.1901 2 arXiv:1509.06429 [pdf, other] On Computational Paths and the Fundamental Groupoid of a Type Arthur F. Ramos, Ruy J. G. B. de Queiroz, Anjolina de Oliveira 3 arXiv:1506.02721 [pdf, other] On the Groupoid Model of Computational Paths Arthur F. Ramos, Ruy J. G. B. de Queiroz, Anjolina G. de Oliveira 4 arXiv:1504.04759 [pdf, other] On the Identity Type as the Type of Computational Paths Ruy de Queiroz (joint work with Anjolina de Oliveira) Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil Propositional Equality, Identity Types and Computational Paths

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